In our paper, we obtain the soliton resolution and long-time asymptotic behavior for the SP equation (REF )
with initial data \(u_0\in H^{1,1}\)  by using \(\bar{\partial }\)  steepest descent method.
This paper is arranged as follows. In section 2, we introduce two kinds of eigenfunctions to
formulate the spectral singularity of the Lax pair for the short-pluse equation. The analytical and asymptotics
of the eigenfunctions are further studied. In section 3, following the
idea in [1]}, we construct a RH problem for \(M(z)\)  to formulate the initial value problem of the short-pluse equation (REF )-() in an
alternative space variable \(y\)  instead of the original space variable \(x\) . In section 4, we introduce
a function \(T(z)\)  to define a new RH problem for \(M^{(1)}(z)\) , which admits a regular discrete spectrum and two triangular decompositions of the jump matrix
near critical point \(\pm z_0\) .
In section 5, by introducing a matrix-valued function \(R(z)\) , we obtain a mixed \(\bar{\partial }\) -RH problem for \(M^{(2)}\)  by continuous extension to \(M^{(1)}\) .
In section 6, we decompose \(M^{(2)}(z)\)  into a
model RHP problem for \(M^{RHP}(z)\)  and a pure \(\bar{\partial }\)  Problem for \(M^{(3)}(z)\) .
The \(M^{RHP}(z)\)  can be obtained via an outer model \(M^{(out)}(z)\)  for the soliton components to be solved in Section ,
and an inner model \(M^{(\pm z_0)}\)  for the stationary phase point \(\pm z_0\)  which are approximated by a solvable model for \(M^{sp}\)  obtained in [1]} in Section .
In section , we compute the error function \(E(z)\)  with a small-norm Riemann-Hilbert problem.
In Section 10, we analyze the \(\bar{\partial }\) -problem for \(M^{(3)}\) .
Finally, in Section 11, based on the result obtained above, a relation formula
is found
\(M(z) = M^{(3)}(z) E(z) M^{out}(z) T(z)^{-\sigma _3},\nonumber \) 
